Gibbons-Hawking Entropy and BMN Strings (2511.08213v1)

Abstract: We provide some up-to-date discussions related to cosmological event horizon and entropy of our universe, then introduce an intriguing idea that there may be a universal finite upper bound for entropy accessible to an observer in consistent theories of quantum gravity. We argue that the BMN (Berenstein-Maldacena-Nastase) strings provide a test of the idea and a possible estimate of the cosmological constant.

• The paper proposes that quantum gravity enforces a universal, finite entropy bound accessible to any observer, as suggested by the Gibbons-Hawking entropy.
• It employs BMN string transitions in a pp-wave background to test how discretized, nonperturbative effects maintain finite entropy even at strong coupling.
• The results imply that operational measurements in string theory may offer a method to compute cosmological constants by linking microphysical entropy with de Sitter space.

Gibbons-Hawking Entropy, BMN Strings, and Universal Entropy Bounds

Overview and Motivation

This work addresses the deep connection between cosmological entropy, specifically the Gibbons-Hawking entropy of de Sitter horizons, and microphysical models from string theory. The central idea is to propose and justify the existence of a universal, finite upper bound on the observable entropy in quantum gravity. The Berenstein-Maldacena-Nastase (BMN) string model in a pp-wave background is proposed as a concrete system in which to test this conjecture, and possibly provide an estimate of the cosmological constant.

Cosmological Entropy in de Sitter Space

The argument is rooted in the standard cosmological model: our universe, accelerating under dark energy, is well-modeled by a positive cosmological constant $\Lambda$. In such spacetime, observers are surrounded by a cosmological event horizon (CEH) at a finite radius, directly mirroring the situation in black hole thermodynamics. The Gibbons-Hawking construction assigns an entropy to this horizon, proportional to its area:

$S = \frac{A}{4 l_p^2}.$

With present cosmological parameters, $S \sim 10^{122}$, vastly exceeding the entropy in any astrophysical subsystems, including the sum over all supermassive black holes, CMB photons, and dark matter relics.

The calculation of the CEH, distinctions between cosmological and particle horizons, as well as arguments for the stability and robustness of the Gibbons-Hawking entropy, are reviewed. These results are insensitive to extra spatial dimensions.

A key feature is that, unlike Bekenstein-Hawking black hole entropy, which has a microscopic counting in string theory for select supersymmetric cases, the microphysical origin of de Sitter entropy remains obscure. Despite intricacies—such as questions about Euclidean path integrals, temperature assignment, and entropic interpretation—the holographic viewpoint supports treating $S_{GH}$ as genuine entropy accessible (in principle) to a single observer.

The Conjecture: A Universal, Observer-Accessible Entropy Bound

The author conjectures that quantum gravity enforces not just context-specific bounds (e.g., Bekenstein bound) but a universal, finite ceiling on the entropy accessible to any observer, regardless of system size, energy density, or process, provided certain reasonable restrictions. This stands in contrast with traditional quantum and quantum statistical systems, where entropy may become arbitrarily large or even infinite due to infinite-dimensional Hilbert spaces, unbounded temperature limits, or the use of continuous spectra and contrived measurement procedures.

Key constraints imposed for this universal bound to remain meaningful include:

• Limiting measurements to natural bases and physically realizable initial states.
• Restricting the number of measurements to a small finite number (excluding repeated, memoryless or unphysical measurements).
• Focusing on type I von Neumann algebras and avoiding ambiguities associated with renormalized trace and entropy in type II or III factors.

In these conditions, the author argues that no robust physical counterexamples are known where entropy surpasses a universal bound in fully quantum gravitational systems.

BMN Strings as a Testbed for the Entropy Bound

To operationalize the conjecture, the tensionless BMN string model in a maximally supersymmetric pp-wave background is investigated. This string theory setup is well-understood, exactly solvable, and is dual (via AdS/CFT) to a particular scaling limit of maximally supersymmetric Yang-Mills theory at large R-charge.

In this background, string states are labeled by discrete quantum numbers (e.g., two string harmonics). Using the AdS/CFT dictionary, transition probabilities between basis states at a given string coupling $g$ (set by $J^2 / N$) can be computed nonperturbatively and correspond to the sum over all genus contributions in the gauge theory. The associated entropy for a single measurement is

$S_m(g) = - \sum_n p_{n,m}(g) \log p_{n,m}(g)$

with $p_{n,m}(g)$ the transition probability (norm squared amplitude).

Key Observations:

• At all finite $g$, $S_m(g)$ is finite, as expected.
• The behavior as $g \to \infty$ (strong coupling, "tensionless" limit) is pivotal. If $p_{n,m}(g)$ becomes uniformly zero for all $n$, the entropy diverges; if not, it remains finite.
• No general principle forces this limit to diverge, and concrete analysis of $p_{n,m}(g)$ for BMN strings is presented as a program for testing the universal bound conjecture.

A proposition formalizes this link: if the transition probabilities vanish uniformly in $n$ as $g \to \infty$, the entropy diverges; otherwise, the entropy remains finite. The necessity of uniform convergence is demonstrated via counterexamples.

Implications and Theoretical Significance

The suggestion that a universal entropy cap exists—emerging naturally in quantum gravity—has implications for the computability of cosmological parameters like $\Lambda$ from microphysical considerations. This approach circumvents the ambiguities inherent to the landscape of string theory vacua because the de Sitter entropy is largely independent of details of low-energy compactification.

If this bound is realized in a model as tractable as BMN strings, it paves the way to:

• Concrete calculations of the upper bound, potentially matching the Gibbons-Hawking entropy and, by inversion, predicting $\Lambda$.
• A shift in how quantum gravity is constrained: via entropy and information, not solely through energy, dynamics, or topology.
• Empirical testability: predictions for cosmological constants that are, in principle, (dis)provable via observation.

In broader context, the result supports the informational/holographic paradigm for quantum gravity and positions entropy/area as a central, physically accessible quantity, not just a formal artifact.

Conclusion

The paper advances the conjecture that quantum gravity enforces a universal, finite entropy ceiling accessible to any observer, substantiated by the analysis of string-theoretic models (BMN strings) and cosmological considerations (de Sitter space). The distinctive feature is operationalizing entropy production through physical, observer-based measurements and string transitions, eschewing unphysical constructions. Testing the behavior of BMN string entropy at strong coupling is presented as a decisive next step.

If successful, this approach would offer a new method for predicting cosmological constants from quantum gravity, independent of detailed model-building, and deepen our understanding of the information-theoretic structure underpinning spacetime in string theory. Future progress hinges on a detailed, nonperturbative analysis of entropy in BMN and related string models, as well as clarifying the precise conditions on measurements and observer-accessible Hilbert space sectors in quantum gravity.